Sewage treatment using AHPD technology
Abstract
Sewage is conventionally treated with AD technology. AHPD technology provides a more efficient alternative. This paper considers replacing AD with AHPD using a two-stage investment model. A real option approach is used to determine the optimal investment thresholds. The findings of this paper show a two-stage investment model is not optimal for this case. A one-stage investment model is preferred. The threshold price of natural gas is slightly higher than the current price of natural gas. This entails that a water sewage treatment plant should postpone investment in an AHPD installation.
Keywords: real option, optimal investment path, sewage treatment
1. Introduction
In the Netherlands increasing circularity of waste cycles is considered a hot topic, and the Dutch government stresses its importance. They argue that the rapidly increasing number of consumers worldwide will result in raw materials become more and more expensive, unless we find a way to deal with our waste cycles in a more efficient manner (RVO, 2019).
Another hot topic is the extraction of natural gas from the natural gas fields of Slochteren, Groningen. As gas extraction leads to local earthquakes, the government decided to lower the extraction to a level that only fulfills the domestic demand. As this level is dropping steadily and alternative sources of energy become available, the Dutch government announced to close the gas extraction from Slochteren in 2030 at its latest (RVO, 2018). Hydrogen is an energy carrier considered to be a possible substitute to natural gas. The amount of research done examining the financial feasibility of implementing hydrogen in our need for energy is overwhelming. However, this technology is at least ten years away from being cost efficient (IEA, 2019), hence most studies come up with negative valuations for hydrogen technologies (Veijer, 2014). As the government aims to close down natural gas production from the gas fields of Slochteren entirely by 2030, a void will be left in the Dutch natural gas market.
Creating methane out of biomass is a possible solution. Methane is the main energy carrier in natural gas. The biological process that creates methane out of biomass is called Anaerobic Digestion (AD). This process happens naturally when microorganisms break down organic matter in environments with little or no oxygen. These circumstances can be recreated in sophisticated installations, and can stimulate the efficiency of disposal of biomass. The resulting product is known as biogas. Such AD installations are mainly used in three industires: agriculture, waste disposal and sewage treatment. In agriculture this process is used for a more efficient disposal of animal waste, such as manure, lowering the excretion of methane directly into the atmosphere. Traditional waste disposal and sewage treatment plants, filter out and burn the wet biomass to dispose it, which requires substantial amounts of energy, and excreting significant amounts of CO2. Newer waste disposal and sewage treatment plants use AD installations to process biomass. As a result, a smaller fraction of the biomass is being burned and less energy is required.
Since AHPD can (partially) fill the void in the natural gas market and increase circularity in the economy it is worth analyzing an optimal investment strategy where an AD biomass processor is replaced with an AHPD equivalent. Furthermore, note that the price of natural gas prices is considered to be highly volatile, and the investment costs are substantial. Therefore, a WSTP would have to be certain this project would not lose its value in the near future. To manage the risk of natural gas price uncertainty, it may be optimal to invest in a sequential manner, where first a sludge processor is installed, and thereafter the OHW processor. Therefore, the following research question is formulated:
How does two-stage investment model affect the optimal investment threshold for an AHPD installation?
The investment model is approached from the perspective of a WSTP. Payoffs are considered to be uncertain and realized upon completion of each investment. Furthermore, the WSTP has the option to postpone the investment, the option life is first considered to be infinite, but later this assumption is removed, investment costs are substantial and the investment is irreversible. The price of natural gas is an important factor in determining the cash flows of the investment stages, which are commonly considered to follow a stochastic distribution. Therefore, this paper sets out to analyze the optimal investment path under constantly growing natural gas prices. Real option theory is implemented to determine the optimal investment decisions, considering the abovementioned assumptions. The Bellman equation is used to analytically determine the optimal investment decision. Since an underlying assumption of the Bellman equation is infinite option lifetime the explicit finite difference method is used to numerically determine the optimal investment decisions, while alleviating this assumption. Both methods will be discussed in section 4 of this paper.
As mentioned, increasing the circularity of the economy is a broadly discussed theme in The Netherlands. This paper is of interest because it analyses the financial competitiveness of a novel technology, which can potentially help solving this social issue. Several papers have evaluated an investment program where investments are done sequentially (see, e.g., Siddiqui and Maribu, 2009; Siddiqui and Fleten, 2010). A stringent assumption in such papers is the unlimited lifetime of the option to invest. It can be argued that when the investment in a novel technology is postponed too long, a more efficient technology will be developed, resulting in the option value considered in this thesis to be valueless. Hence, this paper will explore the implications of this assumption by first assuming unlimited option lifetime, which will thereafter be alleviated. In this way, this paper adds to existing literature.
Chapter five sheds light on the specific values of all variables used as inputs. In chapter six we will analytically and numerically solve the two-staged investment model, discuss the results and perform sensitivity analyses for robustness. Chapter seven concludes this thesis.
2. Background
In this chapter I will first discuss the waste cycle of sludge in The Netherlands. Thereafter, the AHPD system and its process is described.
2.1 Water sewage treatment
In the Netherlands there are 352 WSTPs, owned and operated by 21 different local water authorities (Watersector.nl, 2019). Since not all WSTP’s operate in the same manner, first a more general overview of the operations of a WSTP is given. The operations of a WSTP can be divided into primary, secondary and post treatment. Since AHPD is not involved in the post treatment step, this will not be discussed.
When entering a WSTP, sewage water passes a pump station, where electricity is used to pump the sewage water through the system. Thereafter, the sewage water passes a bar screen, which removes large objects from the water flow, protecting the system from damage and clogs. In some WSTPs a finer screen is placed after the bar screen, called a clog screen, which removes finer solids. After removing solids, the sewage water enters a tank where sewage water is given time to settle. This leads to grit settling at the bottom and grease settling atop of the sewage water, where these can be removed more easily. After grease and grit removal, the output enters a Pre-Settling Tank (PST), which is the last stage in primary treatment. This tank removes part of the organic materials, by scraping the bottom and surface of the tank.
Secondary treatment removes the remaining biomass from the output. During the first stage of secondary treatment, the sewage water enters an Aeration Tank (AT). In the AT multiple processes take place, removing nitrogen compounds and organic dry substance (ODS). These processes require a constant and substantial flow of oxygen, which is an energy intensive process (Unie van Waterschappen, 2014). This entails approximately between 55 and 75% of energy consumption of WSTPs (Stowa, 2018-69). This, in combination with biogenic CO2 emitted from removing ODS the reason for high climate impact of WSTP’s is found (Van Veen, 2020).
From the AT the sewage water is pumped into the Settling Tank (ST), where activated sludge is removed. A large fraction of this sludge enters the sludge treatment phase. After this stage the optional post treatment phase takes place.
burning it in a CHP, but the more modern technique involves using AD to digest the sludge partially prior to transportation. As mentioned before, this process can be done more efficiently with the use of AHPD.
When sewage water first enters the sludge processing stage it contains approximately 4.5% dry substance (DS). Since attempting to burn water is highly inefficient, the sewage water is first dewatered, using a centrifuge, to 24% DS (Van Veen, 2020). Wastewater is then pumped back into the WSTP. After centrifuging, it is optional to dry the sludge even further, usually dependent on the distance from the WSTP to the CHP. The result ranges from 40 to 90% DS. After this stage, the sludge is transported to a CHP. The CHP turns the DS into heat and, dependent on the DS%, electricity (IV-Groep, 2014). Approximately half of WSTP’s process sludge using this method (Van Veen, 2020).
The remaining 50% digests the sludge using AD, prior to transportation to a CHP. Digesting the biomass with AD technology results in the production of biogas. In theory, this gas consists of 50% CO2 and 50% CH4 (methane) molecules. The methane concentration is in practice higher; between 55 and 70% (Lindeboom et al., 2012). CO2 makes up the rest of the gas. Since the biogas does not have the methane concentration of natural gas, it must be upgraded prior to injecting it into the natural gas grid. This process is relatively expensive, since it requires investments in additional equipment, and requires substantial amounts of energy (Lindeboom et al., 2012). Lindeboom et al. (2012) state that it is therefore financially feasible for biogas flows over 100 Nm3/h. Therefore, 75% of produced biogas is currently directly burned in a CHP, while only 25% is upgraded (Van Veen, 2020). After AD treatment, 65% of ODS content remains unprocessed, and enters the same process as WSTP’s not using AD. 2.2 AHPD
3. Literature review
3.1 Real option valuation
3.2 Option to defer investment
When the timing of an investment decision is up to the managers’ discretion, we consider the investment to include the option to defer. In reality, this option exists in almost all investment projects. When a manager decides not to invest in a project at one point in time, the investment does not immediately disappear, but rather remains in existence for at least some time after. When considering the ‘time value of money’ one would consider postponing to have a negative impact on the value of investment. However, real option literature suggests otherwise. For instance, Pennings and Lint (1997) analyzed the value of research and development (R&D) and evaluated the optimal investment choice at every step of the R&D process. One of their findings was that with higher volatility levels the option to defer investment in the next stage increased in value. The idea of value arising from postponing is based on the notion that when the investment is undertaken immediately, the opportunity cost of undertaking this investment later is foregone, and with that, the value of information on market conditions. Therefore, the value of waiting increases with increasing volatility.
3.3 Option to expand
Most investment opportunities are not limited to a one-off investment. When an investment gives rise to another investment opportunity, the first option is considered a compound option (an option on an option). Trigeorgis (1993) evaluates a sequential investment problem where at every stage of the investment process a new option is included. During the building phase the contractor has the option to expand and contract. He finds that at given values of the project, with increasing volatility the option value increases. Pindyck (1988) considers a monopolistic firm with an option of continuous expansion of capacity. He finds that expanding capacity does not occur continuously, but rather in spurs, and is only triggered when demand increases sufficiently above past levels. Both papers consider an investment problem including an option to expand and sequential investment decisions, which is also considered in this thesis.
3.4 Sequential investment
In this paper we analyze the value of sequentially investing in an installation that processes biomass. The investment at stage 1 gives rise to the investment in stage 2, because first the primary task of a WSTP needs to be fulfilled, before expanding capacity. Therefore, the option can be considered a compound option. A vast number of studies have used real option theory to analyze investment problems similar to the problem considered in this thesis.
time-to-complete the investment cost of stage 1 will decrease below that of stage 2, triggering suspension. When there is no option to suspend the investment level of stage 1 never decreases below that of stage 2, since that option value is not included at stage one anymore. Lastly, they found that when time-to-complete is decreased to zero the threshold investment level of stage 1 is always above that of stage 2.
Murto and Nese (2002) analyzed the optimal investment decision between two feedstocks for an electricity plant. The first feedstock considered is fossil fuel, of which the price is uncertain and evolves stochastically. Biomass is considered as an alternative feedstock, with a stable price. The higher the price of natural gas, the less attractive the investment in a gas-fired plant becomes. In this research, Murto and Nese (2002) solve for the threshold price of fossil fuel at which the value of both plants are equal, with the use of the dynamic programming and real option theory. They find that with increasing volatility the value of the gas-fired plant increases, and so does the equilibrium price level at which both plants are equal in value.
Another paper closely related to the analysis done in this paper is that of Siddiqui and Maribu (2009). They implement real option theory to evaluate a multiple stage investment plan for a gas-fired distributed generation unit, where natural gas prices are considered to evolve with uncertainty and stochastically. After the initial investment, the option arises to invest in a heat exchanger and/or invest in expansion of capacity. As there are no cash flows at phase 0, the value at this point exists only of option value. This option value includes the option values of the sequential investments arising from this first investment, and can therefore be considered as a compound option. At phase 1, the microgrid incurs costs for producing, and receives revenues for delivering energy. As cash flows are assumed to be received into perpetuity, value exists of the discounted value of perpetual cash flows and option value to invest in either expansion of production or in the heat exchanger. In the last phase extra costs and revenues are made into perpetuity. At this point there exists no further option to expand this microgrid, and therefore the value is considered to be the discounted cash flows of the base unit plus expansion. For the method, Siddiqui and Maribu (2009) consulted the book of Dixit and Pindyck (1994, p. 321-328). This explains that to find the value of a compound option at phase 0, one must start at phase 2, where no option exists, and work backwards. By finding the value of phase 2, the value of the option at phase 1 can be calculated, resulting in the ability to calculate the value of the option at the initial phase. In their research they found that with higher levels of volatility sequential investment is optimal, whereas with lower low levels of volatility direct investment in all three stages is preferred.
consider a staged process to be optimal for the considered investment problem. At the initial stage no commercialization program exists. At stage 1 commercialization has commenced, thereby decreasing the operating cost of this technology, but no cash flows are received. At the last stage the technology is deployed, cash flows are received and operating costs are considered to continue to decrease. The method of analyzing the thresholds and option values can be considered similar to Siddiqui and Maribu (2009), while differing in the fact that Siddiqui and Fleten (2010) consider two stochastic variables. To be able to find the option values and threshold prices analytically, they create a new variable, by dividing the price of energy by the operational cost, thereby creating a ratio, which can be solved for. This paper helps grasping the effect of coinciding options and how to analyze the interaction of such options in a sequential real option setting.
The papers mentioned above share similarities to this thesis. The model of Siddiqui and Maribu (2009) is similar in the sense that it considers a sequential investment problem with a compound option. The investment decisions are discrete and limited and involve a production function with natural gas as a stochastic variable. The sequential characteristics of this model are due to the threshold prices of natural gas, i.e. when the price drops below a threshold value, the investment in the next stage is warranted. Relating this to my research, the mechanism is such that investment is warranted when the natural gas price rises above a certain threshold value. Murto and Nese (2002) consider a model with competing technologies, where one has stable revenues and the other has uncertain and stochastically evolving revenues. In my research, the starting phase of investment is considered a competing technology with stable revenues. However, since in Murto and Nese (2002) the investment problem is not sequential, but a decision between two technologies, once the investment in the biomass plant has taken place, no option value is included anymore. Therefore, one could consider the model included in this thesis to regard the investment in the plant with stable revenues as already undertaken, currently searching for the threshold value at which it is optimal to switch between technologies.
4. Model
4.1 Compound option
be the first investment stage, since the primary task of a WSTP is processing sludge. Furthermore, the OHW processor cannot be added to AD sludge processor because of the difference in the biological processes occurring within the system. Hence, the AD sludge processor must first be replaced with an AHPD equivalent, before expanding the capacity and adding the OHW processor. When put in option terms: the option to invest in the AHPD sludge processor gives rise to the option to invest in the OHW processor, and can therefore be considered a compound option.
4.2 Geometric Brownian motion
The AHPD installation earns the same revenues as its AD equivalent, plus revenues for producing and selling green gas. Green gas is of comparable quality as natural gas and can be fed into the gas system without further cleaning. Therefore, we consider the price of green gas to be equal to the price of natural gas. The price of natural gas is considered to be uncertain and to evolve stochastically. The stochastic process commonly considered to describe natural gas prices is Geometrical Brownian Motion (GBM). GBM is a basic continuous-time stochastic process, also commonly used in valuating stock options. The price of natural gas, denoted by P, is considered to satisfy the following differential equation:
𝑑𝑃 = 𝛼𝑃 𝑑𝑡 + 𝜎𝑃 𝑑𝑧, (1)
where 𝛼 is a constant representing the expected growth rate of natural gas prices, 𝜎 is the constant volatility of the price process and dz is a the stochastic component with a mean of 0 and variance of 1 per time period. The price is considered to be determined exogenously, since production of one AHPD sludge processor is relatively small in comparison to the total supply in the market, and is therefore considered a price taker.
4.3 The two-stage investment model
Figure 1: A visual representation of the two-staged model. As the price of gas evolves and reaches threshold value P1* an investment decision is warranted. Thereafter, as soon as the price of gas reaches
P2* the last investment decision takes place.
Consider an investment problem where a WSTP has to decide when to invest in a more efficient biomass processor. Figure 1 provides a visual representation of the two-stage investment problem. To receive the full-scale processor the WSTP has to invest Is and Igf for
the sludge processor and OHW processor respectively. At the start, hereafter referred to as stage 0, no investment has taken place, the WSTP receives revenues for processing sludge and they have the option to invest in a more efficient sludge processor now, or wait. As long as the price is below the threshold value of gas prices (discussed into more detail below) the WSTP is considered to be in the waiting regime. As soon as the price of natural gas is equal to its threshold value, denoted by P*, the first investment is made and the WSTP receives 𝜋1, which is the sum of discounted cash flows generated by the operations of the sludge
processor. This phase will be referred to as stage one. In this stage, the WSTP has the option to invest in an OHW processor, thereby increasing the capacity, or postpone investment. Again, as soon as the price of natural gas reaches its threshold value, denoted by P*, the
investment is made and the WSTP receives 𝜋2, which is the incremental sum of discounted cash flows caused by the expansion. At this point, stage 2, there is no investment decision left, since the biomass processor has reached its full scale.
4.3 Analytical solution 4.3.1 Stage 2
To analytically solve this model, we must work backwards. Therefore, stage 2 is considered first. As mentioned, at this point all investments are realized and the biomass processor
operates at its full capacity. No further investment decisions remain, therefore no option value exists. The value of this stage is equal to the sum of discounted cash flows from operations. The value function is derived as follows:
𝑡 𝑃 𝑡 1 𝑃 𝑡 2 𝑃
𝑃1 𝑃2
2(𝑃) = ∫ 𝑋𝑠𝜇𝑠𝑃 (𝛼−𝜏)𝑡𝑑𝑡 − ∫ 𝑋𝑠 𝐼𝑠 −𝜏𝑡𝑑𝑡 3 3 + ∫ 𝑋𝑠𝜇𝑠 𝑠 −𝜏𝑡𝑑𝑡 12 + ∫ 𝑋𝑔𝑓𝜇𝑔𝑓𝑃 (𝛼−𝜏)𝑡𝑑𝑡 − ∫ 𝑋𝑔𝑓 𝐼𝑔𝑓 −𝜏𝑡𝑑𝑡 3 3 + ∫ 𝑋𝑔𝑓𝜇𝑔𝑓 𝑔𝑓 −𝜏𝑡𝑑𝑡 12 (2) => 2(𝑃) = 𝑋𝑠( ( − (𝛼−𝜏)∙3 )𝜇𝑠𝑃 𝜏 − 𝛼 + ( − (−𝜏)∙3 ) 𝐼𝑠+ ( − (−𝜏)∙12)𝜇𝑠 𝑠 𝜏 ) + 𝑋𝑔𝑓( ( − (𝛼−𝜏)∙3 )𝜇𝑔𝑓𝑃 𝜏 − 𝛼 +( − (−𝜏)∙3 ) 𝐼 𝑔𝑓+ ( − (−𝜏)∙12)𝜇𝑔𝑓 𝑔𝑓 𝜏 ) (3) where 𝑋𝑖 is the amount of ODS processed by the relative biomass processor measured in metric tons, 𝜇𝑖 is the efficiency of conversion from biomass to green gas measured in Nm3 per metric ton ODS, P is the price of natural gas, 𝜏 is the discount rate, which is required to be larger than the growth rate of natural gas prices 𝛼, 𝐼𝑖 is the net of all variable costs for the relative biomass processor, 𝑠 and 𝑔𝑓 denote the relative subsidy price and subscript 𝑠 and 𝑓 denote that the variables belong to the sludge processor or OHW processor, respectively. The difference in conversion rates between the two AHPD installations is caused by the difference in composition of the biomass flows. All cash flows are discussed in further detail in section 5.
4.3.2 Stage 1
At stage 1, the WSTP has to decide whether to invest in the expansion of the AHPD sludge processor at investment cost 𝐼𝑔𝑓, or wait. When the price of natural gas (𝑃) is sufficiently high, the WSTP finds that investing in stage 2 immediately is optimal, whereas a lower value for 𝑃 might imply that the investment is postponed to a later point in time. This implies that the value of stage 1 is made up not only of discounted cash flows, but also option value. This option value satisfies the Bellman equation (Dixit and Pindyck, 1994), which is given the following partial differential equation in continuous time:
𝜏 𝑖(𝑃) = 𝜋𝑖(𝑃) + 𝛼𝑃 𝑖′+ 𝜎2𝑃2 𝑖′′
𝛽1= − 𝜎2+ 𝛼 + √( 𝜎2+ 𝛼) 2 + 𝜎2𝜏 𝜎2 > (5) 𝛽2= − 𝜎2+ 𝛼 − √( 𝜎2+ 𝛼)2+ 𝜎2𝜏 𝜎2 < (6)
Plugging the general solution 𝐹1(𝑃) into our Bellman equation we find:
1(𝑃) = 𝜋1(𝑃) + 𝐵1𝑃𝛽1+ 𝐵2𝑃𝛽2, (7) 𝜋1(𝑃) = 𝑋𝑠( ( − (𝛼−𝜏)∙3 )𝜇𝑠𝑃 𝜏 − 𝛼 + ( − (−𝜏)∙3 ) 𝐼𝑠+ ( − (−𝜏)∙12)𝜇𝑠 𝑠 𝜏 ) (8)
To solve for B1 and B2, we have to establish boundary conditions. When the price of natural gas is below the price threshold P* no investment is made and the WSTP earns only its current cash flows. As soon as price threshold 𝑃 is reached, the investment will be made and the increase in cash flows, 𝜋2(𝑃) − 𝜋1(𝑃), is received (𝜋2(𝑃) is equal to V2(P) in the last stage, because all
value comes from discounted cash flows). Now consider, if 𝑃 reaches 0 the value of WSTP should go towards zero (not completely, as revenues are earned which are not influenced by natural gas prices). Because 𝛽2 is smaller than 0, this option value would increase to infinity as P goes to zero, which makes no logical sense. Therefore, we assume 𝐵2 to be equal to 0, resulting in the following value function:
1(𝑃) = 𝜋1(𝑃) + 𝐵1𝑃𝛽1 (9)
To find the value of 𝐵1 and 𝑃 two more boundary conditions have to be created, called the Smooth-Pasting (SP) and Value-Matching (VM) conditions. These conditions are given by:
VM: 1(𝑃𝑔𝑓) = 2(𝑃𝑔𝑓) − 𝐼𝑔𝑓 (10)
SP: 1′(𝑃𝑔𝑓) = 2′(𝑃𝑔𝑓) (11)
VM: 𝑋𝑠( ( − (𝛼−𝜏)∙3 )𝜇𝑠𝑃𝑔𝑓 𝜏 − 𝛼 + ( − (−𝜏)∙3 ) 𝐼𝑠+ ( − (−𝜏)∙12)𝜇𝑠 𝑠 𝜏 ) + 𝐵1𝑃𝑔𝑓 𝛽1 = 𝑋𝑠( ( − (𝛼−𝜏)∙3 )𝜇𝑠𝑃𝑔𝑓 𝜏 − 𝛼 +( − (−𝜏)∙3 ) 𝐼 𝑠+ ( − (−𝜏)∙12)𝜇𝑠 𝑠 𝜏 ) + 𝑋𝑔𝑓( ( − (𝛼−𝜏)∙3 )𝜇𝑔𝑓𝑃𝑔𝑓 𝜏 − 𝛼 +( − (−𝜏)∙3 ) 𝐼 𝑔𝑓+ ( − (−𝜏)∙12)𝜇𝑔𝑓 𝑔𝑓 𝜏 ) − 𝐼𝑔𝑓 (12) SP: (𝑋𝑠 ( − (𝛼−𝜏)∙3 )𝜇𝑠 𝜏 − 𝛼 ) + 𝛽1𝐷1𝑃𝑔𝑓 𝛽1−1 = (𝑋𝑠 ( − (𝛼−𝜏)∙3 )𝜇𝑠 𝜏 − 𝛼 ) + (𝑋𝑔𝑓 ( − (𝛼−𝜏)∙3 )𝜇𝑔𝑓 𝜏 − 𝛼 ) (13)
Combining the VM and SP condition allows us to find the optimal threshold level of 𝑃 at which immediate investment is triggered (𝑃𝑔𝑓):
𝑃𝑔𝑓 = 𝛽1 𝛽1− 𝜏 − 𝛼 ( − (𝛼−𝜏)∙3 )𝜇 𝑔𝑓 (𝐼𝑔𝑓 𝑋𝑔𝑓 −(( − (−𝜏)∙3 ) 𝐼 𝑔𝑓+ ( − (−𝜏)∙12)𝜇𝑔𝑓 𝑔𝑓) 𝜏 ) (14)
Now we can substitute 𝑃𝐺𝐹 into the VM condition and analytically solve for the value of 𝐵1:
𝐵1 = 𝑋𝑔𝑓( ( − (𝛼−𝜏)∙3 )𝜇 𝑔𝑓 𝜏 − 𝛼 ) 𝛽1𝑃 𝛽1−1 (15) 4.3.3 Stage 0
At stage 0 no investment has been made yet and the WSTP has an operational AD sludge processor. This installation is considered to bear no risk, operate into perpetuity and produce a deterministic cash flow, denoted by VIAD. The assumption of a perpetual lifetime is a clear
simplification, as well as riskless returns. However, by making these assumptions the model remains manageable. Furthermore, since we do not know another alternative to AD besides AHPD, no sensible assumption can be made as to when such a sludge processor will be replaced otherwise. At this point the WSTP has to decide whether to invest in an AHPD sludge processor at investment cost 𝐼𝑠, thereby replacing the existing sludge processor. At this stage the WSTP receives revenues equal to 𝜋 which are stable and certain. Furthermore, the value of this stage includes the option value to invest in the sludge processor of stage 1, resulting in the following value function:
𝜋 (𝑃) = 𝑋𝑠( 𝐼𝐴𝐷
𝜏 ),
(17) where 𝐼𝐴𝐷 is the is a capture-all term, which nets all variable costs and incomes. Note that we can find the value of 𝐹 (𝑃) in a similar fashion as in the previous section. The Bellman equation has a similar general solution, given by:
𝐹 (𝑃) = 𝐷1𝑃𝛽1+ 𝐷2𝑃𝛽2 (18)
Except for the denotation of the constants, everything is equal to the conditions of section 4.3.2. By posing similar boundary conditions, 𝐵2 can be set equal to zero and therefore drops out of the equation. Now we can plug in equation (17) and (18) into equation (16) to find the value function, given by:
= 𝑋𝑠( 𝐼𝐴𝐷
𝜏 ) + 𝐷1𝑃
𝛽1 (19)
In order to analytically solve for the threshold level of 𝑃 at which immediate investment is warranted in the AHPD sludge processor, the VM and SP conditions have to be combined again. At the moment 𝑃 = 𝑃 the value of the AD sludge processor plus the option value to invest in stage 1 must be equal to the sum of discounted cash flows of the AHPD sludge processor minus investment costs, plus the option value to invest in stage 2. The VM and SP conditions are given by: VM: 𝑋𝑠( 𝐼𝐴𝐷 𝜏 ) + 𝐵1𝑃𝑠 𝛽 = 𝑋𝑠( ( − (𝛼−𝜏)∙3 )𝜇𝑠𝑃𝑠 𝜏 − 𝛼 +( − (−𝜏)∙3 ) 𝐼 𝑠+ ( − (−𝜏)∙12)𝜇𝑠 𝑠 𝜏 ) + 𝐷1𝑃𝑠 𝛽1 − 𝐼 𝑠 (20) SP: 𝛽1𝐷1𝑃𝑠 𝛽1−1= (𝑋𝑠 ( − (𝛼−𝜏)∙3 )𝜇𝑠 𝜏 − 𝛼 ) + 𝛽1𝐵1𝑃𝑠 𝛽1−1 (21)
Because we already have the value for 𝐵1 we can analytically solve for the constant 𝐷1 and gas price threshold in a similar fashion as section 4.3.2, and are given by:
𝑃𝑠 = 𝛽1 𝛽1− 𝜏 − 𝛼 ( − (𝛼−𝜏)∙3 )𝜇 1 (𝐼𝑠 𝑋𝑠 +( 𝐼𝐴𝐷) − ( − (−𝜏)∙3 ) 𝐼 𝑠− ( − (−𝜏)∙12)𝜇𝑠 𝑠 𝜏 ) (22) 𝐷1= 𝑋𝑠 ( − (𝛼−𝜏)∙3 )𝜇𝑠 𝜏 − 𝛼 𝛽1𝑃𝑠 𝛽1−1 + 𝐵1 (23)
problem and discuss the results of both. The numerical solutions are found using a explicit finite difference model, which will be discussed in the following section.
4.4 Numerical solution
There are several methods to value options with a finite lifetime, such as the tree method, finite differences and Monte Carlo simulation method. In this research the finite differences method is used. Basically, this method tries to approximate the continuous-time differential equation by describing the option movement over time with a set of discrete-time differential equations. This model is very similar to the trinomial tree method. Clewlow and Strickland (1998) describe this method as follows. First, once more consider the price of natural gas to evolve over time following GBM, which is given by:
𝑑𝑃 = 𝛼𝑃 𝑑𝑡 + 𝜎𝑃 𝑑𝑧, (24)
where 𝛼 is the growth rate and 𝜎 is the volatility of natural gas prices. The natural logarithm of price, denoted by 𝑥 is commonly distributed with a constant mean and variance (Clewlow and Strickland, 1998). Then, according to Itô’s lemma, the equation describing the proportional change is given by:
𝑑𝑥 = 𝑣𝑑𝑡 + 𝜎𝑑𝑧, (3. ) (25)
where
𝑣 = 𝛼 − 𝜎2. (3. ) (26)
The price can remain unchanged, or increase or decrease by ∆𝑥 with the probabilities pm, pu and
pd, respectively, over time interval ∆𝑡 (as shown in figure 2). Furthermore, Clewlow and
Strickland (1998) state the time interval is described by:
∆𝑥 = 𝜎√3∆𝑡, (27)
and the movement probabilities are given by: 𝑝𝑢 = (𝜎 2∆𝑡 + 𝑣2∆𝑡2 ∆𝑥2 + 𝑣∆𝑡 ∆𝑥) (28) 𝑝𝑚 = −𝜎 2∆𝑡 + 𝑣2∆𝑡2 ∆𝑥2 (29) 𝑝𝑑 = ( 𝜎2∆𝑡 + 𝑣2∆𝑡2 ∆𝑥2 − 𝑣∆𝑡 ∆𝑥) (30)
When the trinomial process in figure 3.1 is extended a trinomial tree is formed, which is showed in figure 3. Here 𝑖 is the time step and 𝑗 is the price level in relation to the initial price. This means that at node (𝑖, 𝑗) time is 𝑡 = 𝑖∆𝑡 and the asset price 𝑖,𝑗= 𝑗∆𝑥(Zhang, 2018). At the final step 𝑁 the option has reached maturity 𝑇. The value of a call option at maturity is:
𝐶𝑁,𝑗 = ax( , 𝑁,𝑗− 𝐾) . (3. 7)
Figure 3: Trinomial tree
According to Clewlow and Strickland (1998) the option value at forward nodes can be calculated as the discounted expectations, given by:
Figure 4: A trinomial tree in a rectangular grid
However, we cannot calculate the option values at (𝑁𝑇−1, 𝑁𝑗) and (𝑁𝑇−1, −𝑁𝑗) since these nodes are not connected to three nodes in the last column of values. Furthermore, as we continue moving backwards through the grid, with every step the number of nodes it is possible for to calculate the value of decreases with 2. The solution to this problem is to assign reasonable values to the top and bottom nodes each step, thereby creating a fully filled grid.
To determine the reasonable values for the top and bottom nodes boundary conditions need to be set. For a call option, when the stock price goes to 0, the option value should go to 0, as the chance of the option ending up in the money becomes very small. Therefore, all bottom nodes are considered to be 0. When stock prices become very high, the option value will grow proportionally with the stock price. Therefore, the value of the topmost node should be equal to the value of the node below, plus the change in stock prices.
The finite difference method is obtained from the abovementioned model.
5. Data and calibration
In this section the necessary inputs and their derivation are discussed. First the drift rate and volatility of natural gas prices are derived and discussed. Thereafter, the same is done for the discount rate. Lastly, the cash flows of each installation are discussed in more detail.
5.1 Natural gas price factors
parameters for 𝛼 and 𝜎 a method suggested by Hull (2009) was followed. First, time increments need to be added:
𝑑𝑃𝑡 = 𝛼𝑃𝑡 𝑑𝑡 + 𝜎𝑃𝑡 𝑑𝑧𝑡 (32)
From this, natural gas price returns are calculated as follows:
𝑑𝑃𝑡 𝑃𝑡−1
= 𝛼 𝑑𝑡 + 𝜎 𝑑𝑧𝑡, (33)
which follows an Ito process, and can therefore be rewritten as:
𝑙𝑛 ( 𝑃𝑡 𝑃𝑡−1
) = 𝛼 𝑑𝑡 + 𝜎 𝑑𝑧𝑡. (34)
For notational ease I state that:
𝑢𝑖= 𝑙𝑛 ( 𝑃𝑡 𝑃𝑡−1
) (35)
In this case the unbiased estimator of 𝑢𝑖 is given by:
𝑢̅ =
𝑁∑ 𝑢𝑖 𝑁
𝑖=1
(36)
Then the estimator of 𝑢𝑖’s standard deviation is given by:
𝜔 = √ 𝑁 − ∑(𝑢𝑖− 𝑢̅) 2 𝑁 𝑖=1 , (37)
where 𝑢̅ is the unbiased estimator of 𝑢𝑖. Since the collected data is bi-annual a correction must take place to find the yearly volatility of natural gas prices, given by:
𝜎̅ = 𝜔 √𝛾,
(38) Where 𝛾 is the length of periods between returns. Now the annual expected drift rate 𝛼 is given by:
𝛼 =𝑢̅ 𝛾+ 𝜎
2 (4.9) (39)
From this sequence of calculations I have found the following values: 𝜎 = .75% and 𝛼 = .45%.
5.2 The weighted average cost of capital
The relevant discount factor used to calculate the present value of future cash flows is the Weighted Average Cost of Capital (WACC). The WACC is an essential part of analyzing the value of a project since it is used as the risk adjusted discount factor. The formula used in this research comes from Koller et al. (2015) and is given by:
𝑊𝐴𝐶𝐶 =𝐷𝑘𝑑( − 𝑇𝑚) +𝐸𝑘𝑒, (40)
where 𝐷
𝑉 is the level of debt to value, 𝐸
𝑉 is the level of equity to value, 𝑘𝑑 is the cost of debt, 𝑘𝑒 the cost of equity and 𝑇𝑚the project’s marginal tax rate (Koller et al., 2015). The debt and equity to value ratios for a project are most likely not stable. This also goes for the project evaluated in this thesis. Since the project is funded by a certain level of debt, which is paid off over the lifetime of this project the average level of both ratios is used in the WACC calculation. The cost of equity and the cost of debt, generally considered to be the most challenging aspect of WACC calculations, can be calculated using the Capital Asset Pricing Model (CAPM):
𝐸[𝑅𝑖] = 𝑟𝑓+ 𝛽𝑖 𝑀𝑅𝑃, (41)
where E(Ri) is the expected return on a certain asset, rf is the risk free rate, 𝛽𝑖 is the sensitivity of the security to the market and MRP is the market risk premium. The cost of equity equation is given by:
𝑘𝑒 = 𝑟𝑓+ 𝛽𝑒 𝑀𝑅𝑃, (3.3) (42)
where ke is the cost of equity and 𝛽𝑒 is the beta of equity or the sensitivity of a security to the market, regarding equity.
The risk free rate of return is return expected for a riskless investment. In typical times, these are considered to be the bond rates of larger countries, such as the U.S., and in Europe, Germany. However, currently the interest rate on bonds are close to, or even below, zero. In this case, according to Koller et al. (2015), a synthetic risk free rate can be built by adding the expected inflation rate to the long-run average real interest rate on a particular government bond. Koller et al. (2015) continues to state that although the result is different from the actual risk free yield, the low interest rates are an aberration caused by unusual monetary policy, and therefore not a good representation of risk free returns. As soon as the risk free rate returns to historical levels this perspective must be revisited. The long term inflation rate is retrieved from the European Central Bank website (2020), and is estimated to be 1.70%. The average long term interest rate used in this thesis was the 5-year smoothed average 30-year Dutch bond rate, which is retrieved from www.marketwatch.com (2020) and is found to be 0.90%. Therefore the risk-free rate used in this research is 2.60%.
To calculate the beta of equity we must find the sensitivity of this asset to the market returns. This is usually done by regression analysis. However, since this project is not a traded asset, it is not possible to do a regression analysis. An alternative approach to finding the levered beta is given in Koller et al. (2015):
𝛽𝑒 = 𝛽𝑢 + ( − 𝑇𝑚)𝐷
𝐸(𝛽𝑢− 𝛽𝑑), (3.4)
(43) where 𝛽𝑢 is the unlevered beta, or the beta of equity free of debt, 𝛽𝑑 is the beta of debt and
𝐷 𝐸 is the debt to equity ratio. The debt to equity ratio remains stable over the lifetime of this project as debt is payed off over the same period length as the installation is fully depreciated. This ratio was found to be 2.33.
The corporate tax rate is set to be lowered to 21.7% from 2021 onwards (Business.gov.nl, 2020). Therefore, this rate is used throughout this thesis.
resulting in: 𝛽𝑢 = .7. As we now have all necessary elements to calculate the beta of equity, this will be the following step. The result is: 𝛽𝑒 = .65.
Cost of debt is calculated in a comparable fashion:
𝑘𝑑 = 𝑟𝑓+ 𝛽𝑑 𝑀𝑅𝑃, (44)
where kd is the cost of debt and 𝛽𝑑 is the beta of debt. This beta measures the sensitivity of return on debt of the project to the market return. It is calculated by rewriting equation (44):
𝛽𝑑 =𝑘𝑑− 𝑟𝑓 𝑀𝑅𝑃
(45) However, since the beta of debt is needed to calculate the cost of debt, and as seen from the equation (45), the cost of debt is needed to calculate the beta of debt, we are stuck in a loop. Luckily, there is an approximation to (𝑘𝑑− 𝑟𝑓); the credit spread (Koller et al., 2015). The formula for the beta of debt therefore becomes:
𝛽𝑑 = 𝐶 𝑀𝑅𝑃,
(46) where CS is the relevant credit spread. A credit spread is the difference between a corporation’s debt and risk free debt, and is greater for companies with lower credit ratings and higher risk of default (Koller et al., 2015). Damodaran Online (2020) provides a list of approximated credit spreads based on the interest coverage ratio, which is calculated as followed:
𝐼𝐶𝑅𝑡 =
𝐸𝐵𝐼𝑇𝑡 𝐼𝑛𝑡 𝑟 𝑠𝑡 𝐸𝑥𝑝 𝑛𝑠 𝑡
,(3.8) (47)
where EBIT is the earnings before interest and taxes. As only one ICR is needed for the WACC calculation and the ICR rises steeply when debt is close to being fully restituted, the median of ICRs is chosen for finding the relevant approximation for CS. The median ICR equals 6.74, resulting in CS to equal 1.08%, according to Damodaran Online (2020). Therefore: 𝛽𝑑 = . 8. At this point it is possible to calculate the cost of debt and equity and the WACC: 𝑘𝑒 = .56%, 𝑘𝑑 = 3.68% and 𝑊𝐴𝐶𝐶 = 5.78%.
5.3 Cash flows
Because of confidentiality considerations, I was instructed not to mention specific financial parameters of the AHPD installations (For further insights into the financial parameters send an e-mail). Therefore, I will explain the cost items and revenues, and mention the sum of these items. All installation specific variables are provided by Bareau (2020). For notational ease, the subscript 𝑠 and 𝑓 intend to show the variable is specific to the sludge processor and OHW processor, respectively.
common size (K. Zagt., personal communication, April 16, 2020) . Therefore, in consultation with Bareau the WSTP of Emmen was selected as a subject for this case study. This plant processes the sludge of 236.000 p.e., which are projected to produce 3635 metric tons of ODS on a yearly basis, denoted by 𝑋𝑠. The addition of an OHW processor would result in an additional 3635 metric tons of ODS, denoted by 𝑋𝑔𝑓. For processing biomass the WSTP receives four income streams. First are the revenues received for processing the ODS found in sewage water and OHW. The operator receives a GateFee, which is a price received for every metric ton of ODS processed. This is the largest revenue stream.Secondly, the operator receives revenues for every Nm3 gas sold to a gas supplier. The quantity of gas produced can be calculated by multiplying the amount of processed ODS, given by 𝑋𝑠 and 𝑋𝑔𝑓, with the efficiency of conversion, measured by the amount of green gas produced from one metric ton of ODS, denoted by 𝜇𝑠 and 𝜇𝑔𝑓. The price of gas evolves according to the drift rate, which, as mentioned earlier, is denoted by 𝛼. Third, the subsidy revenues are considered. The subsidy income will be received for the first twelve years of operations and is €0.11/Nm3 of green gas produced from sewer sludge, denoted by 𝑠, and €0.19/Nm3 of green gas produced from OHW, denoted by 𝑔𝑓. Lastly, the operator receives revenues for fixating CO2, which is by far the smallest revenue stream.
When considering the costs, the most salient one is of course the investment costs. The remaining cost are variable of nature. Firstly, we consider personnel expenses, which are the salary of three individuals operating the plant. Secondly, the cost of disposing hazardous substances and residual sludge are considered. Thirdly, the cost of energy is considered. Fourth, cost of feeding the gas into the system are incurred. Although the gas is of the same quality as natural gas, it cannot be fed into the system without adding a small amount of odorous gas. This is done with natural gas as well, and prevents gas leaks from going unnoticed. Fifth, we consider the maintenance cost. The last cost factor is a combination of relatively small costs, such as insurance. A sludge processor has an expected lifetime of thirty years.
Table 1: Values of parameters of the base case.
In the numerical model the option is considered to have a finite lifetime. This is set equal to 10 (T=10). The reason for this is that multiple processes are being developed that process biomass, none of which are market-ready at the moment. However, within ten years this will most likely have changed. Therefore, if within ten years the option to invest is not yet exercised, the option expires.
Lastly, the lifetime of the installation is considered to be 30 years(Bareau, 2020).
Chapter 6: Results and Discussion
In this chapter the results of the solution to the Bellman equation are given to our two-stage investment model. Because the underlying assumption of the Bellman equation, the infinite option lifetime, is unrealistic, a numerical model will be solved as well. A numerical programme was used to solve for these values, which is given appendix A. Lastly, since several inputs are estimated with inputs that are relevant today, but may not be any more in the future a sensitivity analysis is performed, to analyze the robustness of the results and the influence of several parameters on the outcome.
6.1 Analytical solution
First, the net present values of each stage will be given. The reason for this is that it demonstrates the difference between the NPV analysis, with its NPV rule, and real option analysis. The net present value of stage 1 is the increase in cash flows between stage 1 and 0, minus the investment costs. At the current price of natural gas, the NPV of stage 1 is -4,573,750 euro. The main reason for this negative results is that the difference in cash flows between both
Parameter Base case value Unit
Drift rate of gas prices (α) 2,45%
Discount factor (τ) 5,78%
Amount of ODS processed by sludge processor (Xs) 3635 Metric tons per year
Amount of ODS processed by OHW processor (Xgf) 3635 Metric tons per year
Variable income of AD sludge processor (VI0) 200 Eur
Variable income of AHPD sludge processor (VI1) 152.5 Eur
Variable income of AHPD OHW processor (VI2) 61.1 Eur
Investment cost of AHPD sludge processor (Is) 9,700,000 Eur
Investment cost of AHPD OHW processor (Igf) 6,350,000 Eur
Efficiency of conversion sludge into green gas (μs) 300 Nm3 green gas per metric ton of ODS
Efficiency of conversion OHW into green gas (μgf) 450 Nm3 green gas per metric ton of ODS
Subsidy for producing green gas out of sludge (Ss) 0.11 Eur per Nm3 green gas
Subsidy for producing green gas out of OHW (Sgf) 0.19 Eur per Nm3 green gas
stages is not large enough to offset the investment costs. The NPV of stage 2 is 12,646,460 euro. The increase in the net cash flows, without considering the sale of gas, alone is almost enough to offset the investment costs.
As real option valuation considers the volatility of gas prices, whereas NPV analysis does not, the expectation is that this method will yield different results. After implementing all parameters into equation (22) the threshold gas price for investing in the AHPD sludge processor, 𝑃𝑠, was found to be 0.782 euro. When implementing this value into the NPV analysis we find a NPV of 2,826,226 euro. This is substantially higher than 0, which is NPV when considering the option value. The threshold gas price for investing in the OHW processor, 𝑃𝑔𝑓, is 0.019 euro. The NPV of stage two at this natural gas price is 89.223 euro, also higher than 0. The difference is caused by the fact that real option analysis considers the volatility of gas prices. Because of the high level of uncertainty, there is value in waiting a bit longer and finding out how the gas price actually evolves. With high uncertainty comes the risk of investing a large sum of money in an installation that produces a substitute of natural gas, while the price of natural gas decreases substantially.
Another noteworthy finding is that the threshold gas price level of stage 1 is higher than that of stage 2. This signals that a sequential investment program might not be optimal for this investment problem. The reason for this is that once the first investment is undertaken, the second is taken immediately as well.
6.2 Numerical solution
6.2.1 Two-stage investment model
than the returns of engaging in the investment,
the NPV, and therefore no investment will be undertaken. When inspecting the base scenario value of sigma in fig. 5, one can observe that the option value is not close to the NPV line. This shows that the option value is relatively large. This is caused by the positive drift rate of natural gas prices. The longer the investor waits, the higher the price of natural price becomes, increasing the value of the investment. At higher levels of sigma, the distance between the NPV line and the option value lines increases. This implies that the more uncertain the price of natural gas is, the more valuable it becomes to postpone the investment, which is in line with the findings of Pindyck (1988) and Murto and Nese (2002). As can be seen in fig. 6 the option value line is close to the NPV line at every value of sigma. This demonstrates that the option to invest in stage 2 holds little value. The main reason for this is mentioned in an earlier section, namely that the income of the OHW processor barring green gas revenues is already close to the investment cost. Hence, there is little value in postponing the investment, as this processor is close to a riskless investment. For this investment stage the influence of sigma is not visible by inspecting the option value lines. It is still visible through the increase in the price thresholds with increasing sigma.
Figure 5: Three option price lines and their respective price thresholds of stage 1 are depicted for volatility levels of 0.107 (base case), 0.207 and 0.307. The black dotted line represents the NPV. Other input is according to base case, see table 1.
Figure 6: Three option price lines and their
respective price thresholds of stage 2 are depicted for volatility levels of 0.107 (base case), 0.207 and 0.307. The black dotted line represents the NPV. Other input is according to base case, see table 1.
Figure 7: The relation between volatility and the price threshold of stage 1. Other inputs are according to base case, see table 1.
The impact of volatility on the threshold price of natural gas is depicted in fig. 7 and 8. Both figures show a non-linear relationship between uncertainty and the threshold price. The impact of the change of volatility on the price threshold increases, which implies that the more uncertain the change in gas price is, the more valuable the option to postpone investment becomes. Practically speaking, this helps determine the impact on the threshold levels, were gas prices to change.
Fig. 9 and 10 demonstrate a scenario in which the discount factor varies between the base case value of 5.8% and increases to 7.8% with steps of 1%. The discount factor influences the option value through its impact on the NPV of the investment. A higher discount factor decreases the payout of engaging in the investment, thus leading to a higher threshold price of natural gas. This effect is on display in both figures. In practice, this helps to determine the impact on the threshold values if any of the underlying assumptions used in estimating the discount factor changes (i.e. risk free rate returning to historic levels would imply a higher discount factor and therefore a higher threshold value).
Figure 9: Three option price lines, NPV lines and the price thresholds of stage 1 are depicted for discount factors of 0.058 (base case), 0.068 and 0.078. Other inputs are according to base case, see table 1.
Figure 10: Three option price lines and their
6.2.2 One-stage investment model
Because the threshold price of natural gas is lower for stage 2 is lower than that of stage 1, a one-stage investment model, where investing in an AHPD installation that can process sludge and OHW at once, is preferred over a two-stage investment model. The reason for this is that the second stage investment adds relatively little risk, caused by the strongly positive NPV. The investment cost, 𝐼𝑓𝑢𝑙𝑙 , is equal to 𝐼𝑠+ 𝐼𝑔𝑓 and the value function is equal to equation 3. The option value, NPV and price threshold are shown in fig. 8. The threshold price for investing is 0.535 euro. This is lower than the threshold price for investing in stage 1 of the two-staged investment model, which warrants invesmtent in both stages simultaneously as well. What does this imply in practice? First, it shows the one-stage investment approach is preferred for the investment problem considered in this thesis. Second, since the threshold price of natural gas is close to the actual price of natural gas, this investment may very well be undertaken in the near future. The influence of volatility on the threshold price is similar to that discussed in the two-stage investment model, a positive non-linear relationship, with increasing impact of volatility on the threshold price.
Chapter 7: Conclusion
This thesis considers an AHPD processor as an alternative to AD sludge processors. AHPD’s biological efficiency is already proven in this regard by Van Veen (2020), leaving the financial feasibility to be analyzed. A WSTP situated in Emmen was used for this case study, at a scale of 237.000 p.e. This investment problem is approached with a two-stage investment model. Because natural gas prices are considered to be highly volatile, the possibility of investing in a sequential manner is considered, as this possibly circumvents the inherent risk of investing in a large AHPD installation. At stage 0 the WSTP operates an AD sludge processor. At stage 1 of this investment model, the WSTP considers investing in an AHPD sludge processor, which
Figure 11: Option price line, NPV line and price threshold are shown for one-stage investment model. Inputs are according to base case, see table 1.
replaces the AD sludge processor. Once the price of natural gas reaches a certain threshold, investment in the AHPD sludge processor is warranted, and we move into the second stage of the investment model. In the second stage, the WSTP considers expanding the AHPD processor to process organic household waste. Using a real option model, the threshold values of the natural gas price are analyzed. At a the base case values of all inputs, the threshold price instituting the stage 1 investment was 1.294 euro per Nm3 of natural gas, and the equivalent value for stage 2 is 0.032 euro. The robustness of these results is tested by varying the value of the discount factor and volatility of gas prices. The findings show that the threshold level triggering investment in both stages increases with volatility, of which the impact increases as well. Furthermore, the relation between the discount factor and threshold prices is also positive. These findings are in line with the characteristics of real option theory, as discussed by Dixit and Pindyck (1994). The results of this thesis proves a two-stage investment approach is not an optimal solution to this investment problem. To demonstrate the two-stage investment approach is not an optimal solution, a one-stage approach is also analyzed. The threshold price of natural gas, warranting investment in both stages at once, is 0.535 euro. This value is not far off the current price of natural gas, implying that in the near future it may be an optimal decision to replace AD sludge processors with an AHPD biomass processor.
There are several limitations to this research. As the variables for volatility and growth rate of gas prices are calculated from bi-annual data, the values may be biased. Another limitation are the inputs collected from Damodaran Online (2020): the credit spread, betas and market risk premium. To calculate these values is considered beyond the scope of this research, yet calculating the actual values will lead to less biased and more reliable results, in particular when considering the discounted cash flows. Lastly, as it proved to be difficult to receive cooperation or any form of data from regional water authorities, the variable costs and revenues of AD sludge processors are not exact values, but rather estimations. To improve the quality of results a more exact value is needed.
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Appendix
Appendix A: The script used to calculate numerical values. import math import numpy as np #Parameters T=10 n=10000 #steps nj=1000 alpha =0.024479 sigma=0.107462
interestrate=0.057849 volatility=0.107462 #dividendrate=alpha
k1=Is - X1*((b*VO1)/ro+(c*mu1*Ss)/ro-(VO0/ro)) k2=Igf - X2*((b*VO2)/ro+(c*mu2*Sgf)/ro)
#Parameters for one-stage investment model
#k1=k1=Is+Igf - X1*((b*VO1)/ro+(c*mu1*Ss)/ro-(VO0/ro))- X2*((b*VO2)/ro+(c*mu2*Sgf)/ro) #X1=7270
#mu1=375
def tri_am( T, P, k1, k2, sigma, ro, X1, X2, alpha, n, nj):
# Calculation of initial parameters dt = T/N dx = sigma*np.sqrt( 3*dt ) nu = alpha- 0.5 * sigma**2 edx=np.exp(dx) pu = (1/2)*dt* ( (sigma/dx)**2 + nu/dx ) pm = 1 - dt*(sigma/dx)**2 - ro*dt pd = (1/2) * dt * ( (sigma/dx)**2 - nu/dx ) const_a1= (X1*mu1*a)/(ro-alpha) const_a2= (X2*mu2*a)/(ro-alpha) # Determination of asset price at maturity
price_level = P * math.exp(-nj * dx) * edx ** np.linspace(0, 2 * nj, 2 * nj + 1)
strike1 = k1 * np.ones(2*nj+1) strike2 = k2 * np.ones(2*nj+1)
payoff1 = np.maximum(0, const_a1 * price_level[0:2*nj+1]-strike1) payoff2 = np.maximum(0, const_a2 * price_level[0:2*nj+1]-strike2) c1 = np.zeros([(n+1),(2*nj+1)])
c2 = np.zeros([(n+1),(2*nj+1)])
# Determination of option value and option value considering early exit c1[n,:] =payoff1[0:2*nj+1]
c2[n,:] =payoff2[0:2*nj+1]
for i in range(n, 0, -1):
c1[i - 1, 1:2 * nj] = pu*c1[i, 2:2 *nj + 1] + pm * c1[i, 1:2 * nj] + pd * c1[i,0:2 * nj - 1] c1[i - 1, 0] = c1[i-1, 1]
c1[i - 1, 2 * nj] = c1[i-1,2*nj-1]+(price_level[2*nj] -price_level[2*nj-1]) c1[i - 1, :]=np.maximum(c1[i-1,:],const_a1*price_level - strike1)
for i in range(n, 0, -1):
c2[i - 1, 1:2 * nj] = pu*c2[i, 2:2 *nj + 1] + pm * c2[i, 1:2 * nj] + pd * c2[i,0:2 * nj - 1] c2[i - 1, 0] = c2[i-1, 1]
c2[i - 1, 2 * nj] = c2[i-1,2*nj-1]+(price_level[2*nj] -price_level[2*nj-1]) c2[i - 1, :]=np.maximum(c2[i-1,:],const_a2*price_level - strike2)
# Determine price threshold P* early_exercise1 = c1[0] > payoff1 i_2 = 0
while early_exercise1[i_2] == True:
P_threshold1 = (price_level[i_2+1] + price_level[i_2])/2 i_2=i_2+1
early_exercise2 = c2[0] > payoff2 i_3 = 0
while early_exercise2[i_3] == True:
P_threshold2 = (price_level[i_3+1] + price_level[i_3])/2 i_3=i_3+1
return (c1[0], c2[0], price_level, (const_a1*price_level[0:2*nj+1]-strike1), (const_a2 * price_level[0:2*nj+1]-strike2), P_threshold1, P_threshold2)
def analytical(k1, k2, ro, sigma, alpha, X1, X2): if sigma ==0: sig_analytic = sigma+0.001 else: sig_analytic=sigma beta_1=(-0.5*sig_analytic**2+alpha+np.sqrt((0.5*sig_analytic**2+alpha)**2+2*(sig_analytic**2)*ro))/sig_an alytic**2
p_threshold1 = (beta_1/(beta_1-1)) * ((ro-alpha) / (a*mu1))*(k1/X1) p_threshold2 = (beta_1/(beta_1-1)) * ((ro-alpha) / (a*mu2))*(k2/X2)
constant_D = ((X2*a*mu2)/(ro-alpha))/((beta_1)*(p_threshold2)**((beta_1)-1))
constant_B = ((X1*a*mu1)/(ro-alpha))/((beta_1)*(p_threshold1)**((beta_1)-1))+constant_D
F_analytic1, F_analytic1_2 =np.zeros(1001), np.zeros(1001) for P in range (1,1000):
if P < p_threshold1:
F_analytic1[P] = constant_B * P ** beta_1 else:
F_analytic2, F_analytic2_2 = np.zeros(1001), np.zeros(1001) for P in range (1,1000):
if P < p_threshold2:
F_analytic2[P] = constant_D * P ** beta_1 else:
F_analytic2[P] = X2 * ((a*mu2*P)/(ro-alpha)+ (b*VO2+c*mu1*Sgf)/ro) - Igf F_analytic2_2[P] = X2 * ((a*mu2*P)/(ro-alpha)+ (b*VO2+c*mu2*Sgf)/ro) - Igf
return (F_analytic1_2, p_threshold1, F_analytic2_2, p_threshold2) # Numerical
y_axis_lattice1, y_axis_lattice2, x_axis, y_axis_2_1, y_axis_2_2, P_star1, P_star2 = tri_am( T, P, k1, k2, sigma, ro, X1, X2, alpha, n, nj)
y_axis_lattice1B, y_axis_lattice2B, x_axisB, y_axis_2_1B, y_axis_2_2B, P_star1B, P_star2B = tri_am( T, P, k1, k2, 0.207, ro, X1, X2, alpha, n, nj)
y_axis_lattice1C, y_axis_lattice2C, x_axisC, y_axis_2_1C, y_axis_2_2C, P_star1C, P_star2C = tri_am( T, P, k1, k2, 0.307, ro, X1, X2, alpha, n, nj)
# Calculates values of analytic option value and threshold
y_axis_analytic1, P_star2_1, y_axis_analytic2, P_star2_2= analytical(k1, k2, ro, sigma, alpha, X1, X2)
y_axis_analytic1B, P_star2_1B, y_axis_analytic2B, P_star2_2B= analytical(k1, k2, ro, 0.207, alpha, X1, X2)
y_axis_analytic1C, P_star2_1C, y_axis_analytic2C, P_star2_2C= analytical(k1, k2, ro, 0.307, alpha, X1, X2)
# Price threshold graphs
print("value_P_when_NPV_=_0", y_axis_2_1 > 0)
print("This is p star lattice", P_star1, "This is p star analytic", P_star2_1)
fig=plt.figure()
ax.set_ylim([-5000000, 40000000]) ax.set_xlim([-0, 2.5])
ax.plot(x_axis, y_axis_lattice1, color='limegreen', lw=2, label='option value (sigma=0.107)')
ax.axvline(x=P_star1, color='limegreen', lw=0.9, linestyle=':', label='threshold (sigma=0.107)')
ax.plot(x_axisB, y_axis_lattice1B, color='orange', lw=2, label='option value (sigma=0.207)') ax.axvline(x=P_star1B, color='orange', lw=0.9, linestyle='--', label='threshold (sigma=0.207)')
ax.plot(x_axisC, y_axis_lattice1C, color='purple', lw=2, label='option value (sigma=0.307)') ax.axvline(x=P_star1C, color='purple', lw=0.9, linestyle='--', label='threshold (sigma=0.307)') ax.plot(x_axis, y_axis_2_1, color='black', linestyle='--', lw=1, label='NPV')
plt.ticklabel_format(style='sci', axis='y', scilimits=(0,0)) ax.set_xlabel("Gas price [P]")
ax.set_ylabel("Option value [F(P)]")
ax.set_title("Option price of stage 1 for sigma=(0.107, 0.207, 0.307)") # Volatility graphs
ax.legend(loc=0)
ax.axhline(y=0, color='blue') t scipy.stats as ss
#import scipy as scip
P_star_graph1 = np.zeros(20) x_axis_sigma = np.zeros(20) for i in range (0, 20):
y_axis_sigma1, y_axis_sigma2, p_level, y_axis_2_sigma1, y_axis_2_sigma2, P_star_graph1[i], P_star_graph2[i] = tri_am( T, P, k1, k2, ((i+4)/50), ro, X1, X2, alpha, n, nj)
x_axis_sigma[i]=(i+4)/50 fig=plt.figure(2)
ax= fig.add_axes([0.11, 0.1, 0.85, 0.82])
p2=np.polyfit(x_axis_sigma, P_star_graph1, 2)
ax.plot(x_axis_sigma, scip.polyval(p2, x_axis_sigma), color='blue')
ax.set_xlabel("Sigma")
ax.set_ylabel("Price threshold [P*]")
